17 votes

Do double integrals always give a volume?

If you think of an ordinary integral $ \int _ a ^ b g ( x ) \, \mathrm d x $ as an area (or a signed area in case $ g $ might take negative values), then you can think of a double integral $ \int _ R ...
Toby Bartels's user avatar
  • 4,714
15 votes
Accepted

Show $\int_0^{\pi/2}\int_0^1 e^{t+t^{\tan\theta}}dtd\theta=\frac{\pi}{4}(e^2-1)$

First begin with the substitution $s=\tan\theta\implies \frac{1}{1+s^2}ds=d\theta$, giving us \begin{align*}I&:=\int_{0}^{\frac{\pi}{2}}\int_{0}^{1}e^{t+t^{\tan\theta}}\, dt\, d\theta \\ &\,=\...
KStarGamer's user avatar
  • 5,099
12 votes

Interpreting a notation in calculus of variations (differentiating with respect to a derivative)

Consider a functional $J[y]$ defined by: $$J[y] = \int_a^b F(x, y, y') dx \tag{1}$$ I think it will be helpful to remind you right away that your notation implies that you have already agreed to ...
hft's user avatar
  • 556
10 votes

A tricky system of non-linear multivariate equations

Add the first equation to twice the second and to the third and express the result as $$ (x+a)^2+(y+b)^2=(z+c)^2 $$ Draw the following triangles to illustrate this equation as well as the equations $$...
John Wayland Bales's user avatar
9 votes

Inequality $\sum_{k=1}^{n} \frac{\log(a_k)}{1+a_{k}^{2}} \leqslant 0$

We have $$ 2 \sum_{k=1}^{n} \frac{\log(a_k)}{1+a_{k}^{2}} = \sum_{k=1}^{n} \left( \frac{2\log(a_k)}{1+a_{k}^{2}} - \log(a_k)\right) \\ = \sum_{k=1}^{n}\frac{(1-a_k^2)\log(a_k)}{1+a_{k}^{2}} \le 0 $$...
Martin R's user avatar
  • 113k
9 votes
Accepted

Do double integrals always give a volume?

The topic of double integrals $$\iint_D f(x,y) \ dxdy$$ affirms that from the geometric point of view it can be interpreted as the volume of the solid obtained by extending the plane region $D$ until ...
Sebastiano's user avatar
  • 7,649
8 votes
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Find limit using generalized binomial theorem.

You have exactly the right idea! As a life-pro-tip, one common way to simplify estimates like these is to try to get all of our norms to look similar. Here we have a $|x+y|$ in the numerator (which ...
Chris Grossack's user avatar
8 votes
Accepted

How to evaluate $\displaystyle \int_{-\pi/2}^{\pi/2} f(x) dx$ where $f(x)=\cos(x)+\sin(f(x))$

$\def\J{\operatorname J} \def\H{\operatorname H} $ Apply the Kepler equation Fourier series, so Bessel J appears $$\text{Area}=\int_{-\frac\pi2}^\frac\pi2\cos(x)+\sum_{n=1}^\infty\frac{2\sin(n\cos(x))}...
Тyma Gaidash's user avatar
8 votes
Accepted

Matrix Derivative of $X^3$

Since matrix multiplication is not commutative, you have to be more careful and more explicit. Compute the derivative in the direction of a tangent vector (i.e., matrix) $H$: \begin{align*} Df_A(H) &...
Ted Shifrin's user avatar
8 votes
Accepted

Integrate $\int_0^1\left(\int_0^\pi \frac{u}{\sqrt{1+u^2-2u \cos\phi}} d\phi\right)du$

With $K$ as the Complete Elliptic Integral of the First Kind with Paramater $k$, $$K:=K(k)=\int_0^{\pi/2}\frac{1}{\sqrt{1-k^2\sin^2t}}dt$$ Also using Landen's Transformation: $$K\left(\frac{2\sqrt{x}}{...
Miracle Invoker's user avatar
8 votes
Accepted

How to calculate $\int _0^1 \int _0^1\left(\frac{1}{1-xy} \ln (1-x)\ln (1-y)\right) \,dxdy$

$$ \begin{align} &\hspace{5mm}\int _0^1 \int _0^1 \frac{\ln (1-x)\ln (1-y) }{1-xy} \,dxdy \\&=\int_0^1 \ln(1-x)\frac{Li_2(\frac x{1-x})}{x}dx =\int_0^1 {Li_2(\frac x{1-x})}d[Li_2(1)-Li_2(x)]\\ ...
Quanto's user avatar
  • 97.7k
7 votes
Accepted

A clean way to identify $\nabla(|f|^2)$ with $2f\overline{f'}$ for a holomorphic $f$.

If you identify $\nabla W$ with $\frac{\partial}{\partial x}W + i \frac{\partial}{\partial y}W $ then you can use the Wirtinger derivatives $$ {\begin{aligned}{\frac {\partial }{\partial z}}&={\...
Martin R's user avatar
  • 113k
7 votes

vector-valued function visualization

You just draw the vector $(x^2-y^2,2xy)$ at position $(x,y)$:
Michael Hoppe's user avatar
6 votes
Accepted

Asymptotic behavior of the inverse Fourier transform of $\frac{1}{|k|^2 + 1}$

$$f(x)=\int_{-\infty}^\infty...\int_{-\infty}^\infty\frac {e^{ik_1x_1+ik_2x_2+...+ik_nx_n}}{1+k_1^2+...+k_n^2}dk_1...dk_n$$ $$=\int_{-\infty}^\infty...\int_{-\infty}^\infty e^{ik_1x_1+ik_2x_2+...+...
Svyatoslav's user avatar
  • 15.7k
6 votes

How to check if the limit exists in multivariable calculus

You can use the estimate $$ \bigg| y e^{-\frac{1}{\sqrt{x^2 + y^2}}}\bigg| \le \sqrt{x^2 + y^2} \cdot e^{-\frac{1}{\sqrt{x^2 + y^2}}}, $$ switch to polar coordinates and compute the limit $$ \lim_{r \...
Virtuoz's user avatar
  • 3,656
6 votes
Accepted

How to check if the limit exists in multivariable calculus

You idea is perfectly fine and I think that hints or suggestions given are a little bit confusing. Indeed squezee theorem and polar coordinates are really not necessary in this case. We can simply ...
user's user avatar
  • 155k
6 votes

How do I evaluate $\int_{0}^{\infty}{e^{-x}\frac{\sin^2{x}}{x}dx} = \frac{\ln{5}}{4}$ using iterated integrals?

Amusingly, I solved this as part of a homework assignment last semester. The hint that I was given would probably prove helpful though: verify and use the fact that $$ \int_0^1 e^{-y} \sin(2xy) \, \...
PrincessEev's user avatar
6 votes
Accepted

The directional derivative equals dot product of gradient and a unit vector. But what if the function is not totally differentiable?

I think most of what needs to be said is in the comments, but in an attempt to set things out clearly: If $f\colon \Omega\to \mathbb R$ is a function defined on an open set $\Omega$ of $\mathbb R^n$ (...
krm2233's user avatar
  • 4,568
6 votes

Root test application question (usefulness of root test for the series $\sum 1/n^n$)

Ratio test does work: Let $a_n=1/n^n$ and it follows $$\left|\frac{a_{n+1}}{a_n}\right|=\frac{n^n}{(n+1)^{n+1}}\leq\frac{n^{n+1}}{(n+1)^{n+1}}=\left(1-\frac1{n+1}\right)^{n+1}\to\frac1e\quad\text{as }...
Mengchun Zhang's user avatar
6 votes

Using infinitesimals in multivariable calculus

There are pitfalls with using notation that is too abbreviated when working in multivariate calculus, whether or not one uses infinitesimals. Partial derivatives are treated carefully in Keisler's ...
Mikhail Katz's user avatar
  • 42.6k
6 votes

Minimizing $ \sum\limits_{cyc}\frac{\sqrt{5a+8bc}}{8a+5bc}$ with $ \sum\limits_{cyc}ab=1$

Some thoughts. Let $$x := 5a + 8bc, \quad y := 5b + 8ca, \quad z := 5c + 8ab,$$ $$u := 8a + 5bc, \quad v := 8b + 5ca, \quad w := 8c + 5ab.$$ Let $$A := \frac{2\sqrt{2} - \sqrt{5}}{3}, \quad B := \frac{...
River Li's user avatar
  • 37.8k
6 votes

Calculating Volume of "fat sphere" defined by $x^8+y^8+z^8 = 64$

The technique is due to Dirichlet. It is spelled out fairly completely in the first edition of Whittaker and Watson, but reduced to a homework exercise by the fifth. Delighted to find that the 1902 ...
Will Jagy's user avatar
  • 140k
6 votes
Accepted

Finding the Inverse of a Function on the Sphere

Your transformation can be written under a matrix form : $$\pmatrix{x'\\y'\\z'}=\pmatrix{\cos(z) & \sin(z) & 0\\ \sin(z) & -\cos(z) & 0\\0&0&1}\pmatrix{x\\y\\z}$$ Its inverse ...
Jean Marie's user avatar
6 votes

Evaluating a double integral involving a biquadratic with variable coefficients

Integrate w.r.t. $u$ first; the denominator can be rearranged as $$\left(1035 t^4 + 2070 t^2 + 1035\right) u^2 + \left(729 t^4 + 594 t^2 + 121\right) = 1035 \left(t^2+1\right)^2 u^2 + \left(27t^2+11\...
user170231's user avatar
  • 19.6k
6 votes
Accepted

How to integrate $\int_{0}^{1} \int_{0}^{1} \ln\left(\frac{1}{\sinh^2(x) + \cosh^2(y)}\right) \,dx\,dy$

$$ \begin{split} J &= \int \frac{\ln(y(x^2+1)+x(y^2+1))}{xy}\, dx\\ & = \frac{1}{y} \int \frac{\ln(y(x^2+1)+x(y^2+1))}{x}\, dx \\ & = \frac{1}{y} \int \frac{\ln(x+y)+\ln(x+\frac{1}{y})+\ln(...
kingking's user avatar
  • 108
6 votes
Accepted

How to integrate $\int_{0}^{1} \int_{0}^{1} \tanh^{-1}\left(\frac{x}{y} + \frac{y}{x}\right) \,dx\,dy$

Hint: Per symmetry $$\int_{0}^{1} \int_{0}^{1} \tanh^{-1}\left(\frac{x}{y} + \frac{y}{x}\right) dxdy= 2\int_{0}^{1} \int_{0}^{y} \tanh^{-1}\left(\frac{x}{y} + \frac{y}{x}\right) \,dxdy $$ Then, ...
Quanto's user avatar
  • 97.7k
6 votes
Accepted

Derivative of $Ax x^\top A$ with respect to $x$

Let's look at pertubations $$f(x+v) = A(x+v)(x+v)^TA = Axx^TA + Axv^TA + Avx^TA + Avv^TA$$ The derivate is often defined as the unique linear function such that: $$f(x+v) = f(x) + D_{f;x}(v) + \...
Snake707's user avatar
  • 1,041
5 votes
Accepted

Is $\frac{\partial}{\partial x}(x-y)$ ill-defined?

The only way in which $\frac{\partial}{\partial x}(x-y)$ is ill-defined, is the same way in which the function $x-y$ is ill-defined. That is, the very instant we define the function $x-y$, its ...
Brian Moehring's user avatar

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